I am trying to solve the book "Linear Algebra " written by S H Friedberg, where it is given:
Prove $AB$ is invertible implies both $A $ and $B $ are both invertible ($A $ and $B $ both are square matrices).
My try----$AB $ is invertible so $\det (AB) $ is not $0$ implies $\det (A) $ and $\det (B)$ both are nonzero. So both $A $ and $B $ are invertible. But that book did not say the idea of determinant of matrix before that problem.So I believe we can prove that result not using determinant, but how?